Measuring an angle in non-inertial frame reference

In class, we came up with an equation, a=gtanθ, to calculate the acceleration of a car using a hanging pendulum in Earth's frame of reference (θ being the angle that the pendulum makes with the vertical when the car accelerates). So far, I know that that the pendulum moves in the opposite direction of the car's acceleration and only tension force and gravity are acting on the pendulum in inertial frame of reference.
so eq1. Tsinθ=ma
eq2. Tcosθ=mg and when you solve for a, you get a=gtanθ

But we are trying to develop procedures for an experiment that verifies this equation. We need to find both acceleration and angle that the pendulum makes in an accelerating cart and prove that the equation is valid. We have a motion sensor to detect the acceleration of a moving cart. We are making the cart to accelerate at a constant rate by hanging a mass and dropping it. The part that we are not sure about is how to measure the angle that the pendulum makes with the vertical in the accelerating cart. We thought about taping a protractor to the cart and taking snapshots during the acceleration, but it's difficult to get a clear picture. Do you have any ideas or suggestions to improve our experiment?

I would imagine that if the pendulum is initially hanging vertically when you release the cart, then the pendulum will swing back and overshoot the angle θo that you are trying to measure, thus making the measurement of θo difficult.

In order to avoid this, you might try using a support on the cart (shown in blue in the figure) to hold the pendulum at some initial angle before letting the cart go. If the initial supporting angle is less than θo, then the ball will lose contact with the support after releasing the cart. Thus, by trial and error, you could find the position of the support where the pendulum just starts to lose contact with the support when the cart is released. Maybe this would help. I have never tried it, so it is just a suggestion.

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I would imagine that if the pendulum is initially hanging vertically when you release the cart, then the pendulum will swing back and overshoot the angle θo that you are trying to measure, thus making the measurement of θo difficult.

In order to avoid this, you might try using a support on the cart (shown in blue in the figure) to hold the pendulum at some initial angle before letting the cart go. If the initial supporting angle is less than θo, then the ball will lose contact with the support after releasing the cart. Thus, by trial and error, you could find the position of the support where the pendulum just starts to lose contact with the support when the cart is released. Maybe this would help. I have never tried it, so it is just a suggestion.

Thank you for the suggestion, it's definitely worth trying! But I need to write lab procedures that would allow other people to repeat the same experiment. I don't know if trial and error is what they want..